1. Decision Making Under Uncertainty

Decision Making Under Uncertainty

Introduction to Decision Making Under Uncertainty

Every day, individuals, businesses, scientists, and governments make decisions without knowing exactly what will happen.

For example:

• Should a company invest in a new product?

• Is it better to repair an old machine or purchase a new one?

• How much stock should a shop order when customer demand is uncertain?

• Should a farmer plant a crop that produces higher profits but is more vulnerable to drought?

• Which transportation option offers the best balance between cost, reliability, and travel time?

These decisions involve uncertainty, because several outcomes are possible and the final result cannot be predicted with certainty.

Mathematics provides methods for comparing these alternatives.

One of the most important tools is expected value, which combines possible outcomes with their probabilities to calculate a probability-weighted average.

However, a good decision may also depend on risk, the severity of possible losses, available resources, and the consequences of different outcomes.

Decision making under uncertainty involves using probability, expected value, and mathematical reasoning to compare alternatives and select an appropriate course of action.

Understanding Uncertainty and Risk

Although the words risk and uncertainty are often used together, they have slightly different meanings in decision analysis.

Risk describes situations where different outcomes are possible and their probabilities can be estimated.

Uncertainty is a broader term describing situations where the future is unknown. In some cases, even the probabilities of different outcomes may be difficult to estimate.

Examples

In many mathematical problems, probabilities are provided or can be estimated from data.

These probabilities allow us to calculate expected outcomes.

Expected Value

The expected value of a random variable is the probability-weighted average of its possible outcomes.

For a discrete random variable X:

E(X) = Σ[x × P(X = x)]

where:

• E(X) is the expected value.

• x is a possible numerical outcome.

• P(X = x) is the probability of that outcome.

• Σ means to add the products for all possible outcomes.

Interpretation

Expected value represents the long-run average outcome if the same decision or random experiment were repeated many times under the same conditions.

It does not necessarily represent an outcome that will actually occur.

For example, the expected number of heads in three fair coin tosses is 1.5.

However, obtaining exactly 1.5 heads is impossible.

Expected value is an average, not a guaranteed result.

Worked Example: Expected Profit

Suppose a company is considering launching a small product.

The possible profits are:

A negative profit represents a loss.

Step 1: Multiply Each Outcome by Its Probability

High sales:

0.30 × $20,000 = $6,000

Moderate sales:

0.50 × $8,000 = $4,000

Low sales:

0.20 × (−$5,000) = −$1,000

Step 2: Add the Results

E(X) = $6,000 + $4,000 − $1,000

E(X) = $9,000

Interpretation

The expected profit is $9,000.

This does not mean the company will earn exactly $9,000.

The actual outcome could be a $20,000 profit, an $8,000 profit, or a $5,000 loss.

The expected value provides a mathematical summary of the possible outcomes and their probabilities.

Expected Monetary Value

When outcomes are measured in money, expected value is often called expected monetary value (EMV).

EMV = Σ[Probability × Monetary outcome]

EMV can be used to compare:

• Investment opportunities.

• Business strategies.

• Equipment purchases.

• Insurance decisions.

• Product launches.

• Project alternatives.

If the objective is to maximize expected monetary gain, the strategy with the highest EMV is preferred.

However, maximizing EMV does not always produce the most suitable decision when financial losses or other consequences are significant.

Comparing Alternative Strategies

Suppose a company is choosing between two business strategies.

Strategy A: Conservative Expansion

Expected value:

EMV(A) = 0.80 × $12,000 + 0.20 × (−$2,000)

= $9,600 − $400

EMV(A) = $9,200

Strategy B: Aggressive Expansion

Expected value:

EMV(B) = 0.40 × $30,000 + 0.60 × (−$4,000)

= $12,000 − $2,400

EMV(B) = $9,600

Comparing the Results

Strategy B has the higher expected profit.

However, it also has a higher probability of loss and a larger potential loss.

A company focused on maximizing expected profit may prefer Strategy B.

A company concerned about financial stability may prefer Strategy A.

The mathematically preferred decision depends on the objective and the constraints of the decision-maker.

Comparing Risks and Rewards

Expected value is important, but it does not fully describe risk.

Two strategies may have the same expected value while producing very different outcomes.

Example

Consider two options.

Option A

Receive $500 with certainty.

Expected value:

E(A) = $500

Option B

• 50% probability of receiving $1,200.

• 50% probability of losing $200.

Expected value:

E(B) = 0.50 × $1,200 + 0.50 × (−$200)

= $600 − $100

E(B) = $500

Both options have the same expected value.

However, Option A guarantees $500, while Option B involves substantial uncertainty.

Interpretation

A risk-averse person may prefer Option A.

A person willing to accept uncertainty for the possibility of a larger reward may prefer Option B.

Neither preference is automatically mathematically incorrect.

Expected value is only one part of the decision.

Variance and Standard Deviation in Decision Making

The variance of a random variable measures how widely possible outcomes are spread around the expected value.

For a discrete random variable:

Var(X) = Σ[(x − μ)² × P(X = x)]

where:

μ = E(X)

The standard deviation is:

σ = √Var(X)

A larger standard deviation indicates greater variability in possible outcomes.

Worked Example: Comparing Two Options

Consider the previous options.

Option A guarantees $500.

Therefore:

E(A) = $500

Var(A) = 0

σ(A) = 0

Option B offers:

• $1,200 with probability 0.5.

• −$200 with probability 0.5.

Its expected value is $500.

Calculate the variance:

Var(B) = 0.5 × (1200 − 500)² + 0.5 × (−200 − 500)²

= 0.5 × 700² + 0.5 × (−700)²

= 245,000 + 245,000

Var(B) = 490,000

Standard deviation:

σ(B) = √490000

σ(B) = $700

Interpretation

Both options have the same expected value, but Option B has a much larger standard deviation.

This means its possible outcomes vary more widely around the expected value.

Standard deviation provides a useful measure of risk when comparing outcomes on the same scale.

Decision Tables

A decision table, also called a payoff table, organizes possible decisions and their outcomes under different conditions.

Suppose a shop must decide how much stock to order.

The shop has two options:

• Order 100 units.

• Order 200 units.

Customer demand may be:

• Low

• High

The profits are:

Suppose:

P(low demand) = 0.40

P(high demand) = 0.60

Expected Value of Ordering 100 Units

EMV(100) = 0.40 × $1,000 + 0.60 × $1,000

EMV(100) = $1,000

Expected Value of Ordering 200 Units

EMV(200) = 0.40 × (−$500) + 0.60 × $2,500

= −$200 + $1,500

EMV(200) = $1,300

Decision

Ordering 200 units has the higher expected profit.

However, ordering 100 units avoids the possibility of a loss under the two scenarios.

The appropriate decision depends on whether maximizing expected profit or limiting financial risk is more important.

Decision Trees

A decision tree is a diagram showing possible decisions, uncertain events, and their consequences.

Decision trees are useful when a problem involves several stages.

Two common types of nodes are:

• Decision nodes: Points where a choice is made, often represented by squares.

• Chance nodes: Points where an uncertain outcome occurs, often represented by circles.

Example: Business Expansion

A company must decide whether to expand.

If it expands:

• There is a 60% probability of earning $20,000.

• There is a 40% probability of losing $5,000.

If it does not expand:

- The profit is $6,000.

Step 1: Calculate the Expected Value of Expanding

EMV(expand) = 0.60 × $20,000 + 0.40 × (−$5,000)

= $12,000 − $2,000

EMV(expand) = $10,000

Step 2: Calculate the Value of Not Expanding

EMV(do not expand) = $6,000

Step 3: Compare

Expanding has the higher expected monetary value.

Decision

If the company wants to maximize expected profit and can tolerate the possible loss, expanding is the preferred strategy.

If avoiding losses is essential, not expanding may be more appropriate.

Analysing Decisions with Several Stages

Some decisions involve a sequence of uncertain events.

For example, a company might:

1. Invest in product development.

2. Conduct a market test.

3. Decide whether to launch the product.

4. Experience high or low customer demand.

Decision trees can represent these stages.

Solving Multi-Stage Decision Trees

A common method is rollback analysis.

Start at the final outcomes and work backward.

At each chance node:

Calculate the probability-weighted expected value.

At each decision node:

Choose the alternative with the preferred value according to the decision criterion.

Continue backward until reaching the initial decision.

This allows complex decisions to be evaluated systematically.

Risk Attitudes

People and organizations differ in how they respond to uncertain outcomes.

Three common attitudes toward risk are:

Risk-Averse

A risk-averse decision-maker prefers greater certainty when alternatives have the same expected value.

For example, they may prefer a guaranteed $500 over a risky option with an expected value of $500.

Risk-Neutral

A risk-neutral decision-maker evaluates alternatives primarily according to expected monetary value.

If two options have the same expected value, they are equally attractive under this criterion.

Risk-Seeking

A risk-seeking decision-maker may prefer an uncertain option over a guaranteed option with the same expected value.

The possibility of a large reward may be especially attractive.

Important Principle

Risk attitudes influence decisions, but they do not change the mathematical expected values of the alternatives.

Expected Utility

Expected monetary value is not always enough to describe preferences.

For example, losing $10,000 may have a very different impact on a small business than on a large corporation.

Utility is a mathematical representation of how desirable an outcome is to a decision-maker.

Expected utility is calculated using:

E[U(X)] = Σ[P(X = x) × U(x)]

where:

U(x) represents the utility of outcome x.

A decision-maker can compare alternatives using expected utility rather than expected money alone.

Example

A person may value financial security more highly than the possibility of earning a much larger amount.

A utility function can represent that preference.

Expected utility provides a more flexible framework for decision-making when risk attitudes matter.

The Value of Information

Sometimes a decision-maker can obtain additional information before making a decision.

For example:

• A company might conduct market research.

• A farmer might purchase a detailed weather forecast.

• A manufacturer might inspect equipment.

• A doctor might request an additional diagnostic test.

Information can improve decisions by reducing uncertainty.

Expected Value of Perfect Information

The expected value of perfect information (EVPI) measures how much better the expected outcome could be if the decision-maker knew the future state before choosing.

EVPI = Expected value with perfect information − Best expected value without perfect information

EVPI represents the maximum amount a risk-neutral decision-maker should be willing to pay for perfect information under the model.

Worked Example: Value of Market Research

Consider the earlier stock-ordering problem.

Probabilities:

P(low demand) = 0.40

P(high demand) = 0.60

The best expected value without additional information is:

$1,300

With Perfect Information

If the shop knew demand would be low, it would order 100 units and earn $1,000.

If it knew demand would be high, it would order 200 units and earn $2,500.

Expected value with perfect information:

EV(with perfect information) = 0.40 × $1,000 + 0.60 × $2,500

= $400 + $1,500

= $1,900

Calculate EVPI

EVPI = $1,900 − $1,300

EVPI = $600

Interpretation

Perfect information would increase expected profit by $600.

Therefore, under this model, a risk-neutral shop owner should not pay more than $600 for perfect information.

Real market research is imperfect, so its expected value would generally be lower.

Decisions When Probabilities Are Unknown

Sometimes probabilities cannot be estimated reliably.

In these situations, expected value may not be appropriate.

Alternative decision criteria can be used.

Maximax Criterion

Choose the strategy with the highest possible payoff.

This is an optimistic approach.

Maximin Criterion

Choose the strategy with the highest worst-case payoff.

This is a conservative approach.

Minimax Regret Criterion

Choose the strategy that minimizes the largest possible regret.

Regret is the difference between the best payoff that could have been obtained in a particular state and the payoff from the chosen strategy.

Example

Maximax: Strategy B, because its maximum payoff is $20,000.

Maximin: Strategy A, because its minimum payoff is $5,000, higher than the minimum payoffs of the other strategies.

For minimax regret:

• Under low demand, the best payoff is $5,000.

• Under high demand, the best payoff is $20,000.

Minimax regret: Strategy C, because its maximum regret is the smallest.

Different decision criteria can produce different choices.

Sensitivity Analysis in Decision Making

Decision models depend on estimated probabilities.

If these probabilities change, the preferred strategy may also change.

Sensitivity analysis examines how changes in assumptions affect the decision.

Worked Example

Suppose a company is considering a project.

If successful, the project earns $15,000.

If unsuccessful, it loses $5,000.

Let p be the probability of success.

Expected value:

EMV = 15000p − 5000(1 − p)

Expand:

EMV = 15000p − 5000 + 5000p

EMV = 20000p − 5000

Find the Break-Even Probability

Set expected value equal to zero:

20000p − 5000 = 0

20000p = 5000

p = 0.25

Interpretation

• If p > 0.25, the expected profit is positive.

• If p = 0.25, the expected profit is zero.

• If p < 0.25, the expected profit is negative.

The break-even success probability is 25%.

This result helps the company evaluate whether its estimate of the probability of success is sufficiently high.

Comparing Strategies Across Different Probabilities

Using the previous example:

EMV = 20000p − 5000